{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Probability formulas\n",
    "\n",
    "Let's explore the formulas used in the project and also visualize the impact of even tiny bias over a number can have over the played tickets. This part is not necessary to understand the TensorFlow 2.0 part, feel free to skip to next section."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "from matplotlib.ticker import PercentFormatter\n",
    "import itertools\n",
    "from collections import defaultdict\n",
    "import math\n",
    "import matplotlib\n",
    "%matplotlib inline\n",
    "matplotlib.rcParams.update({'font.size': 18})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Base distribution: which numbers are preferred\n",
    "\n",
    "Our goal is to find the players bias towards each number using only the lottery results and the number of players that won. To understand how the two are connected, we can start from a given base distribution, showing how likely players are to choose a given number from 1 to 49 and another independent distribution of how inclined they feel between 1 and 10 (called the \"lucky\" number). Our final question is then: how likelly are players to choose the given winner numbers?\n",
    "\n",
    "Let's build the formula step-by-step. If you're impatient, you can skip to the last section of this page.\n",
    "\n",
    "In this file, we'll consider a fake preference distribution. Since we're talking about probability and all possible options are known, we must have:\n",
    "\n",
    "$$ \\sum_{i} B_i = 1 $$\n",
    "\n",
    "In the graphs below, the dashed black represent what an uniform, that is unbiased, distribution would look like."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1008x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Most common is 1.7x more likelly than least common\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1008x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Most common is 1.9x more likelly than least common\n"
     ]
    }
   ],
   "source": [
    "np.random.seed(17)\n",
    "probs1 = np.exp(np.random.normal(0, 0.1, 49))\n",
    "probs1 = pd.Series(probs1 / np.sum(probs1), index=range(1, 50))\n",
    "probs2 = np.exp(np.random.normal(0, 0.3, 10))\n",
    "probs2 = pd.Series(probs2 / np.sum(probs2), index=range(1, 11))\n",
    "\n",
    "probs1.sort_values(ascending=False).plot.bar(figsize=(14, 5), color='C0', width=0.5)\n",
    "plt.axhline(1 / 49, color='black', linestyle='--')\n",
    "plt.gca().get_yaxis().set_major_formatter(PercentFormatter(1))\n",
    "plt.title('Probability of choosing a given ball')\n",
    "plt.show()\n",
    "print('Most common is {:.1f}x more likelly than least common'.format(probs1.max() / probs1.min()))\n",
    "\n",
    "probs2.sort_values(ascending=False).plot.bar(figsize=(14, 5), color='C1', width=0.5)\n",
    "plt.axhline(1 / 10, color='black', linestyle='--')\n",
    "plt.gca().get_yaxis().set_major_formatter(PercentFormatter(1))\n",
    "plt.title('Probability of choosing a lucky ball')\n",
    "plt.show()\n",
    "print('Most common is {:.1f}x more likelly than least common'.format(probs2.max() / probs2.min()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Taking pairs\n",
    "\n",
    "What is the likelihood of players choosing a given pair of numbers, say 12 and 17? Let's ignore the second set of numbers for the moment.\n",
    "\n",
    "Rather surprisingly, it's **not** $ b_{12} * b_{17} $! And why not? Because:\n",
    "\n",
    "1. after choosing 12 as their first number, the second (17) will be drawn from the remaining **48** numbers. It means, its likelihood is now $ b_{17} / (1 - b_{12}) $, since 12 is no longer in the pool. You can convince yourself if you take the extreme case of a lottery with only two numbers from which players have to pick two. Do you see how the second number will certainly be picked?\n",
    "\n",
    "2. besides, a player can either pick 12 and then 17 or do it in the reverse order, the two cases should be added\n",
    "\n",
    "That revised expression is:\n",
    "\n",
    "$$ P(a, b) = a \\frac{b}{1 - a} + b \\frac{a}{1 - b} $$\n",
    "\n",
    "The graph below ilustrate the most and least likely pairs from our example distribution. Note how the 1.7 times preference over a single number grows into 2.6x for pairs. This tendancy will be accentuaded as we grow the size of the set of chosen numbers."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1008x360 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Most common is 2.6x more likelly than least common\n"
     ]
    }
   ],
   "source": [
    "def exact_comb_prob(probs):\n",
    "    prob = 0\n",
    "    for perm in itertools.permutations(probs):\n",
    "        term = perm[0]\n",
    "        remainder = 1 - term\n",
    "        for p in perm[1:]:\n",
    "            term *= p / remainder\n",
    "            remainder -= p\n",
    "        prob += term\n",
    "    return prob\n",
    "\n",
    "def exact_comb_distro(probs, n_balls):\n",
    "    distro = {}\n",
    "    for comb in itertools.combinations(sorted(probs.index), n_balls):\n",
    "        distro[', '.join(map(str, comb))] = exact_comb_prob(probs.loc[list(comb)])\n",
    "    return pd.Series(distro)\n",
    "\n",
    "def plot_comb_distro(distro, n_top=15):\n",
    "    plt.figure(figsize=(14, 5))\n",
    "    plt.subplot(121)\n",
    "    distro = distro.sort_values(ascending=False)\n",
    "    distro.head(n_top).plot.bar()\n",
    "    plt.axhline(1 / len(distro), color='black', linestyle='--')\n",
    "    plt.gca().get_yaxis().set_major_formatter(PercentFormatter(1))\n",
    "    \n",
    "    plt.subplot(122, sharey=plt.gca())\n",
    "    distro.tail(n_top).plot.bar()\n",
    "    plt.axhline(1 / len(distro), color='black', linestyle='--')\n",
    "    plt.tight_layout()\n",
    "\n",
    "probs1_2 = exact_comb_distro(probs1, 2)\n",
    "plot_comb_distro(probs1_2)\n",
    "plt.suptitle('Probability of choosing two given balls (top and bottom 15)', y=1)\n",
    "plt.show()\n",
    "print('Most common is {:.1f}x more likelly than least common'.format(probs1_2.max() / probs1_2.min()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Take it to the three\n",
    "\n",
    "The reasoning is mostly the same when compared to taking pairs, however there are 6 ways of taking three numbers, so our formula will have 6 terms. Look how the denominator decreases within each term.\n",
    "\n",
    "$$ P(a, b, c) = a \\frac{b}{1 - a} \\frac{c}{1 - a - b} +\n",
    "a \\frac{c}{1 - a} \\frac{b}{1 - a - c} +\n",
    "b \\frac{a}{1 - b} \\frac{c}{1 - b - a} + \\\\\n",
    "b \\frac{c}{1 - b} \\frac{a}{1 - b - c} +\n",
    "c \\frac{a}{1 - c} \\frac{b}{1 - c - a} +\n",
    "c \\frac{b}{1 - c} \\frac{a}{1 - c - b} $$\n",
    "\n",
    "You can already where we're reading to for our full 5-numbers set. There are 120 permutations of 5 numbers, so our final equation will have 120 adding terms, which one composed of 5 multipling terms."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1008x360 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Most common is 3.8x more likelly than least common\n"
     ]
    }
   ],
   "source": [
    "probs1_3 = exact_comb_distro(probs1, 3)\n",
    "plot_comb_distro(probs1_3)\n",
    "plt.suptitle('Probability of choosing three given balls (top and bottom 15)', y=1)\n",
    "plt.show()\n",
    "print('Most common is {:.1f}x more likelly than least common'.format(probs1_3.max() / probs1_3.min()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Simplifying a bit\n",
    "\n",
    "Unfortunately, the exact equation above will have too many terms for our case of 5-numbers sets (we're still ignoring the lucky number). 120 terms may seen doable and we'll should test it, however we should remember the optimized will derivate this gigantic expression for each of the 49 variables (the base probabilities we're looking for).\n",
    "\n",
    "In fact, the main concern is not computation time. Our dataset is rather small and we could compensate it with different techniques. My concern is numerical stability! With a formula with *many* adding and multipling similar terms, the errors in the float operations can add up and sometimes just give straight up unacceptable results. I don't understand enough about the issue in fact, it's more of a fear.\n",
    "\n",
    "The idea is to cut corners and simplify our equation. We'd be losing precision for performance and stability. Deal!\n",
    "\n",
    "The bad thing about our formula is that each factor is different, making it hard to squeeze all them together. What if we replaced the denominator with something more predictable, like the average of the arguments?\n",
    "\n",
    "From the 3-numbers set above, see how all denominators will be factored out (here $\\rho$ is the average of a, b, c):\n",
    "\n",
    "$$ P(a, b, c) = \\frac{6 a b c}{(1 - \\rho) (1 - 2 \\rho)} $$\n",
    "\n",
    "It's a lot simpler in fact. So much so, we can now write our full 5-number-set formula:\n",
    "\n",
    "$$ P(a, b, c, d, e) = \\frac{120 a b c d e}{(1 - \\rho) (1 - 2 \\rho) (1 - 3 \\rho) (1 - 4 \\rho)} $$\n",
    "\n",
    "Below an example of the difference between the exact and this simplified formula. It's precise to the 4th significant digit:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.000071151%\n",
      "0.000071136%\n"
     ]
    }
   ],
   "source": [
    "def simple_prob(probs, numbers):\n",
    "    term = 1\n",
    "    remaing_prob = 1\n",
    "    avg = probs[numbers].mean()\n",
    "    for number in numbers:\n",
    "        term *= probs[number] / remaing_prob\n",
    "        remaing_prob -= avg\n",
    "    return math.factorial(len(numbers)) * term\n",
    "\n",
    "top_3_bottom_2 = [42, 11, 6, 2, 40]\n",
    "exact = exact_comb_prob(probs1[top_3_bottom_2])\n",
    "simple = simple_prob(probs1, top_3_bottom_2)\n",
    "print('{:.9%}'.format(exact))\n",
    "print('{:.9%}'.format(simple))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# I'm feeling lucky\n",
    "\n",
    "We've been ignoring the lucky number up to this point. Fortunately, it's straightforward: since selecting the lucky number $l$ from 1 to 10 is an independent process from the other 5 numbers, we just have to multiply the previous formula by its likelihood. We'll call it $P_1$ for reasons that will be evident in the next section:\n",
    "\n",
    "$$ P_1(a, b, c, d, e, l) = \\frac{120 a b c d e l}{(1 - \\rho) (1 - 2 \\rho) (1 - 3 \\rho) (1 - 4 \\rho)} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The winner does not take it all\n",
    "\n",
    "We've focused only in the jackpot: getting all 5 and the lucky number correct. But players can also smaller prizes with smaller points: the French loterry has nine prize levels in total as of today. However, up until 2017 they had only six. Therefore, we'll use those six prize levels so that we can use historical data from 2008 until today:\n",
    "\n",
    "1. (Jackpot) 5 numbers + lucky\n",
    "2. 5 numbers and no lucky\n",
    "3. exactly 4 numbers\n",
    "4. exactly 3 numbers\n",
    "5. exactly 2 numbers\n",
    "6. exactly 1 number + lucky or no number + lucky\n",
    "\n",
    "Notice how these six levels are mutually exclusive, that is, a player can win at most at one of them.\n",
    "\n",
    "The probability of the second level can be derived from $P_1$. See, our player has to get all 5 numbers and NOT the lucky one:\n",
    "\n",
    "$$ P_2(a, b, c, d, e, l) = \\frac{120 a b c d e (1 - l)}{(1 - \\rho) (1 - 2 \\rho) (1 - 3 \\rho) (1 - 4 \\rho)} $$\n",
    "\n",
    "The third level will require more effort. See, all 49 numbers are divided into two groups: the 5 \"good\" numbers and 44 \"bad\" ones. Our player has to choose exactly 4 good and 1 bad (the lucky number doesn't matter). There are 5 way of selecting 4 out of 5:\n",
    "\n",
    "$$ P_3(a, b, c, d, e) = Q_3(a, b, c, d) + Q_3(a, b, c, e) + Q_3(a, b, d, e) + Q_3(a, c, d, e) + Q_3(b, c, d, e) $$\n",
    "\n",
    "However there are 44 ways of selecting the bad number. Here, we'll apply a similar logic as before to simplify things a bit: we'll assume all bad numbers have the same likelihood. This is really important, since for the other levels, the number of combinations explodes! At level 6, there will be 1,086,008 ways of doing so!\n",
    "\n",
    "$$ Q_3(a, b, c, d) = 44 \\frac{120 a b c d \\tau}{(1 - \\rho) (1 - 2 \\rho) (1 - 3 \\rho) (1 - 4 \\rho)} $$\n",
    "\n",
    "Where $\\tau$ is the average over the bad numbers and $\\rho = (a+b+c+d+\\tau)/5$. Notice how that's a bit counter intuitive: the aim of this project is to to find how choices are biased and now we've just assumed there are not!\n",
    "\n",
    "Numerically, though, the difference is very small, on the 4th significant digit on the example below:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.012745845%\n",
      "0.012738320%\n"
     ]
    }
   ],
   "source": [
    "def exact_p3(good_probs, bad_probs):\n",
    "    prob = 0\n",
    "    for gp in itertools.combinations(good_probs, 4):\n",
    "        for bp in bad_probs:\n",
    "            prob += exact_comb_prob([*gp, bp])\n",
    "    return prob\n",
    "\n",
    "def simple_p3(good_probs, bad_probs):\n",
    "    prob = 0\n",
    "    tau = np.mean(bad_probs)\n",
    "    for gp in itertools.combinations(good_probs, 4):\n",
    "        rho = np.mean([*gp, tau])\n",
    "        prob += 44 * 120 * np.product(gp) * tau / ((1 - rho) * (1 - 2*rho) * (1 - 3*rho) * (1 - 4*rho))\n",
    "    return prob\n",
    "\n",
    "good_numbers = [3, 14, 15, 35, 32]\n",
    "good_probs = probs1[good_numbers]\n",
    "bad_probs = probs1[np.setdiff1d(probs1.index, good_numbers)]\n",
    "print('{:.9%}'.format(exact_p3(good_probs, bad_probs)))\n",
    "print('{:.9%}'.format(simple_p3(good_probs, bad_probs)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# I'm feeling (not so) lucky\n",
    "\n",
    "We can proceed identically for level 4, now with 10 ways to choose 3 good numbers:\n",
    "\n",
    "$$ P_4(a, b, c, d, e) = Q_4(a, b, c) + Q_4(a, b, d) + Q_4(a, b, e) + Q_4(a, c, d) + Q_4(a, c, e) + Q_4(a, d, e) + Q_4(b, c, d) + Q_4(b, c, e) + Q_4(b, d, e) + Q_4(c, d, e) $$\n",
    "\n",
    "Each of which with 946 ways to select 2 bad numbers, using the nice simplification above and $\\rho = (a+b+c+2\\tau)/5$:\n",
    "\n",
    "$$ Q_4(a, b, c) = 946 \\frac{120 a b c \\tau^2}{(1 - \\rho) (1 - 2 \\rho) (1 - 3 \\rho) (1 - 4 \\rho)} $$\n",
    "\n",
    "One more time, with 10 ways to pick 2 good numbers:\n",
    "\n",
    "$$ P_5(a, b, c, d, e) = Q_5(a, b) + Q_5(a, c) + Q_5(a, d) + Q_5(a, e) + Q_5(b, c) + Q_5(b, d) + Q_5(b, e) + Q_5(c, d) + Q_5(c, e) + Q_5(d, e) $$\n",
    "\n",
    "And 13244 ways of picking 3 bad ones:\n",
    "\n",
    "$$ Q_5(a, b) = 13244 \\frac{120 a b \\tau^3}{(1 - \\rho) (1 - 2 \\rho) (1 - 3 \\rho) (1 - 4 \\rho)} $$\n",
    "\n",
    "The sixth and final level will break out of this routine a bit with the reappearance of the lucky number:\n",
    "\n",
    "$$ P_6(a, b, c, d, e, l) = Q_6(a, l) + Q_6(b, l) + Q_6(c, l) + Q_6(d, l) + Q_6(e, l) + R_6(l) $$\n",
    "\n",
    "With $Q_6$ choosing 4 bad numbers and the poor $R_6$ getting all 5 wrong!\n",
    "\n",
    "$$ Q_6(a, l) = 135751 \\frac{120 a l \\tau^4}{(1 - \\rho) (1 - 2 \\rho) (1 - 3 \\rho) (1 - 4 \\rho)} $$\n",
    "$$ R_6(l) = 1086008 \\frac{120 l \\tau^5}{(1 - \\rho) (1 - 2 \\rho) (1 - 3 \\rho) (1 - 4 \\rho)} $$"
   ]
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